Properties Of Functionally Graded Material Research Articles

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments

This paper aims to analyze the combined influences of pore imperfection, initial geometric imperfection, elasticity of tangential constraints of ends, elastic foundations and elevated temperature on the nonlinear vibration of functionally graded material (FGM) beams. The pores are assumed to distribute into FGM according to even and uneven patterns and effective properties of porous FGM are estimated using a modified rule of mixture. Motion equations of geometrically imperfect beams are established within the framework of Timoshenko beam theory incorporating von Kármán nonlinearity and foundation interaction. Analytical solutions are assumed to satisfy simply supported and clamped conditions of beam ends and Galerkin method is applied to derive a nonlinear time ordinary differential equation. The frequencies of nonlinear free vibration are determined employing fourth-order Runge-Kutta numerical integration scheme. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies. The results reveal that frequency nonlinearity is more significant for beams with tangentially restrained ends, especially at elevated temperatures. It is also found that geometric imperfection increases the linear frequency and lowers ratio of nonlinear-to-linear frequencies.

Nonlinear thermomechanical vibration of initially stressed functionally graded plates with porosities

AbstractThe large‐amplitude vibration of initially stressed functionally graded material (FGM) rectangular plates with porosities is investigated in this paper. All edges of the plate are simply supported and uncompressed edges are tangentially restrained. Initial stresses are caused by in‐plane compressive load acting on freely movable edges or thermally induced forces at restrained edges. Pores present in FGM through even and uneven distributions. Material properties are assumed to be temperature‐dependent and, due to the presence of the pores, the effective properties of FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on the classical plate theory taking into account von Kármán nonlinearity and initial geometric imperfection. These equations are solved using approximate analytical solutions along with the Galerkin method to obtain a time differential equation containing quadratic and cubic nonlinear terms. Nonlinear frequencies are determined by means of numerical integration employing the fourth‐order Runge–Kutta scheme. Parametric studies are carried out to analyze the effects of material and geometry properties, porosity volume fraction and distribution, pre‐existent compressive and thermal loads, imperfection and degree of tangential constraints of unloaded edges on the natural frequencies and nonlinear to linear frequency ratio of FGM plates.

Vibration and thermomechanical transient response of doubly curved FGM panels with porosities and elastically restrained edges

The present paper aims to analyze the influences of porosities, elevated temperature and elasticity of in-plane boundary conditions on the free vibration and nonlinear transient response of functionally graded material (FGM) doubly curved panels subjected to uniform lateral pressure. Porosities are assumed to exist in the FGM according to even and uneven distributions. Effective properties of porous FGM are determined using a modified rule of mixture. Basic equations in terms of deflection and stress function are derived basing on the classical shell theory taking into account von Kármán–Donnell nonlinearity and initial geometrical imperfection. Analytical solutions for simply supported panels are assumed and Galerkin method along with Runge–Kutta scheme are applied to obtain the deflection–time response for nonlinear dynamic analysis. Numerical results demonstrate that tangential constraints of edges have extremely significant influences on the vibration and transient response of porous FGM panels exposed to an elevated temperature. The analysis also reveals that porosities have negative and positive effects on the dynamic response of FGM panels at reference and elevated temperatures, respectively. Furthermore, when the edges are restrained, initial imperfection importantly influence the behavior tendency of porous FGM panels exposed to preexist thermal environment.

Voxel-based design and characterization of functionally graded materials

Functionally graded materials (FGM) have numerous merits that are appealing to various industrial sectors. Great advancements have been made in FGM manufacturing technologies during the past few decades, especially in voxel-based 3D printing in recent years. Nevertheless, the development of FGM design and characterization methods is considerably lagging behind. The existing FGM design methods are mostly geometry based, and the characterization of FGM properties mainly relies on the conventional micromechanical models; both have fundamental limitations that severely hinder the wide application of FGMs in industries. In this study, two types of voxel-based approaches, i.e., micromechanics-based and statistics-based, are developed for FGM design and characterization. Four representative FGMs are designed and characterized using the proposed approaches. The study results show that FGM properties are dependent on the resolution of FGM image for both the approches, particularly if the phase materials have large contrast in their properties. For the micromechanics-based methods, FGM properties are also dependent on the micromechanics formula that is used in the determination of voxel effective properties. The study shows that the statistics-based approach is more reliable and more accurate. With the new voxel-based 3D printers generally available to the industry, the proposed voxel-based design and characterization approaches will have wide applications in various industrial sectors such as aerospace, energy, biomedical and electrical engineering.

Nonlinear dynamic snap-through and vibrations of temperature-dependent FGM deep arch under sudden thermal shock

In this paper, nonlinear thermally induced vibrations of deep arches made of functionally graded material (FGM) are investigated. The two surfaces of the arch have different thermal conditions, one surface is under thermal shock and the other is maintained at the reference temperature. The properties of FGM depend on temperature. Using the Crank–Nicolson and the central finite difference methods, the one-dimensional heat conduction equation is solved through the thickness of the arch, and the temperature profile can be obtained according to the uncoupled thermoelasticity theory. The equations of motion of the arch are extracted by combining the linear thermoelastic equation, assumptions of uncoupled thermoelasticity laws, and kinematic assumptions of a deep arch with geometrical nonlinearity. The equations of motion are discretized by using the Ritz method with polynomial shape functions and then solved with the help of the β−Newmark time marching scheme and Newton–Raphson linearization technique. The effects of various parameters such as opening angle, slenderness ratio, and arch power law index have been analyzed. Moreover, the Budiansky and other criteria are used to obtain the dynamic buckling temperature.

Analytical Solution for the Three-Dimensional Vibration of a Rectangular Functionally Graded Material Plate with Two Simply Supported Opposite Faces

Analytical solutions based on three-dimensional (3D) elasticity for the vibrations of functionally graded material (FGM) plates are valuable for assessing the validity and accuracy of various plate theories and numerical approaches. Few benchmark 3D analytical solutions for the vibrations of FGM plates are available in the literature. In this study, analytical solutions based on Fourier series and 3D elasticity were developed for the first time for the vibrations of FGM rectangular plates with two simply supported opposite edge faces. The distributions of the properties of FGMs through the thickness follow a simple power law. The proposed solutions were validated by conducting comprehensive convergence studies on the vibration frequencies of square plates with different thickness-to-side ratios and boundary conditions as well as comparisons with published results. The benchmark nondimensional frequencies were tabulated for plates with free boundary conditions on the top and bottom faces and six combinations of boundary conditions on the other two faces. Moreover, the effects of aspect ratio and gradient index on the vibration frequencies of FGM plates were investigated. The influence of the thickness ratio of the FGM layer to the homogenous layer on the vibration frequencies of sandwich plates with FGM face sheets and a homogeneous core was also studied.

Deflection analysis of functionally graded equal strength beams

In this study, equal strength cantilever and simply supported beams made of functionally graded material (FGM) whose material properties vary though the height direction were investigated. These equal strength cantilever FGM beams were loaded with uniformly distributed load and a point load at the tip and simply supported FGM beams were loaded with uniformly distributed loads. They have all variable cross-section and straight axis. For calculating equivalent material properties of FGMs, power law distribution and Mori-Tanaka model were used. A computer program was developed for the analysis of the problem. The dimensionless deflection values for cantilever beams and simply supported beams were obtained for different materials by the help of developed computer program. Obtained results are presented in the form of tables and graphs which may be useful for the researchers.

Thermo-torsional buckling and postbuckling of thin FGM cylindrical shells with porosities and tangentially restrained edges

This paper aims to analyze the effects of porosities, tangential constraints of boundary edges and elevated temperature on the buckling and postbuckling behaviors of thin functionally graded cylindrical shells subjected to uniform torsion. The porosities are assumed to exist within functionally graded material (FGM) according to even and uneven distributions. The properties of constituents are assumed to be temperature dependent and effective properties of porous FGM are determined using a modified rule of mixture. Mathematical formulations are established within the framework of classical shell theory taking into account von Kármán–Donnell nonlinearity and elasticity of tangential edge constraints. Multi-term analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is used to determine buckling torsional load and load–deflection relation for postbuckling analysis. Numerical results show that porosities have remarkably deteriorative influences on the torsional buckling resistance and postbuckling load capacities of FGM cylindrical shells. This study also reveals that tangential edge constraints have pronouncedly beneficial and detrimental effects on the postbuckling strength of torsion-loaded porous FGM cylindrical shells at room and elevated temperatures, respectively.

BENDING PROPERTIES OF A FUNCTIONALLY GRADED POLYMERIC COMPOSITE REINFORCED WITH A HYBRID NANOMATERIAL

In this paper, functionally graded polymer hybrid nanocomposites have been produced by silica (SiO2) nanoparticles and alumina (Al2O3) nanoparticles distributed in a matrix of epoxy during the ultra-sonication via hand lay-up method. The variation in nanoparticles volume fraction (Vf.) has been given in the thickness direction for reaching the gradation. Each layer has a thickness of 1.2 mm through various concentrations of nanoparticles and is sequentially cast in acrylic moulds to fabricate the graded composite sheet with a 6 mm thickness. To fabricate the functionally graded layers, various concentrations of different nanoparticles (1.5% SiO2, 1% SiO2, epoxy, 2% Al2O3 and 3% Al2O3) have been used for tensile and compressive testing each isotropic layer of functionally graded material (FGM). The mechanical property that was studied for pure epoxy, isotropic and FGM was the flexural resistance. The flexural properties of FGM, isotropic nanocomposite (1% SiO2 + 2% Al2O3) and pristine epoxy, for evaluating their mechanical properties, including flexural stress-strain criteria and flexural Young's modulus, were determined via a Three-point bending test, with loading from the side of silica and alumina for the hybrid-FGM and at one side for the isotropic hybrid nanocomposite and pristine epoxy. The mechanical properties (tensile and compression) and the density of every layer were obtained for the epoxy resin and nanocomposites. They can benefit from the Finite Element Analysis (FEA) of the Three-point bending test via the Design Modeler (ANSYS workbench). The results of experiments were confirmed via building a detailed 3D FE model. Also, the advanced deformation results from the FE model were found in good agreement with the experimental outcomes.

Manufacturing and Measuring Mechanical Properties of Continuous Functionally Graded Beam

In this study, glass-filled epoxy functionally graded material (FGM) was prepared by adopting the hand lay-up method. The vertical gravity casting was used to produce a continuous variation in elastic properties. A 30 % volume fraction of glass ingredients that have mean diameter 90 μm was spread in epoxy resin (ρ = 1050 kg/m3). The mechanical properties of FGM were evaluated according to ASTM D638. Experimental results showed that a gradually relationship between Young’s modulus and volume fraction of glass particles, where the value of Young’s modulus at high concentration of glass particles was greater than that at low concentration, while the value of Poisson’s ratio at high concentration of glass particles was lower than that at low concentration. The manufacture of this FG beam is particularly important and useful in order to benefit from it in the field of various fracture tests under dynamic or cyclic loads.

A convenient method for the strength evaluation of cracked functionally graded material plates under different loadings

As functionally graded material (FGM) plates are widely used in engineering and industrial fields, the fracture strength evaluation of the cracked FGM plate becomes increasingly important. However, due to the gradation of the mechanical properties of FGM, few analytical methods are simple and available. In this paper, a convenient strength evaluation method for the cracked FGM plates under different loadings is proposed in terms of the stress intensity factor (SIF) appearing at a crack tip. This method uses the stable relationship of singular stress fields between FGM and homogeneous material to greatly simplify the complex SIF calculation process of the cracked FGM plate. The results show that the SIFs of FGM plates under tension, bending, and three-point bending (experiment) can be obtained conveniently and accurately by using two simple formulas, and the trends of critical SIF can also be investigated directly even if the SIF solution of the homogeneous plate (reference problem) is unknown. Comparisons with the published theoretical and experimental results indicated that the present method is reliable, precise, and practical to determine SIF and perform strength prediction.

Analysis of the modal frequency of a functionally graded cylindrical shell

The modal frequencies of a functionally graded material (FGM) cylindrical shell with different geometry dimensions and material constituents are studied under eight boundary conditions. The main goal is to reveal the influencing mechanism of different factors on the modal frequencies of the FGM shell. Firstly, the Voigt model is used to describe the properties of FGM. Secondly, the motion equation of the FGM shell is derived using Hamilton’s principle. Thirdly, the modal frequency equation of the FGM shell is obtained based on a trigonometric function for the eight boundary conditions. Lastly, the effects of geometry dimension, material constituents, and boundary conditions on the modal frequencies of the FGM shell are analyzed through case studies. Results indicate that the modal frequencies obtained are good agreement with those from previous literature and the finite element method. The geometry dimensions of the FGM shell, the material constituents of the FGM, and the boundary conditions are crucial to the modal frequencies. This work can serve as a reference for the design of FGM and optimization of the structure of FGM shells.

Transient response of collinear Griffith cracks in a functionally graded strip bonded between dissimilar elastic strips under shear impact loading

This article analyses the interaction between a central and two symmetrically placed collinear Griffith cracks subjected to transient response under anti-plane shear impact loading. The cracks are situated in a strip constituted by functionally graded material (FGM) bonded between two dissimilar elastic strips of equal thickness. The material properties of FGM are assumed to vary exponentially as a function of thickness. Applying integral transforms, the boundary value problem reduces to a system of singular integral equations in the Laplace transformed domain. These equations are solved numerically using the Lobatto-Chebyshev collocation quadrature approach. The inverse Laplace transform is used to find the approximate expressions of dynamic stress intensity factors (DSIFs). The striking feature of the article is the study of phenomenal changes of shielding and amplification through dynamic stress magnification factors (DSMFs) at the tips of the cracks under the sudden impact loading applied at the upper material surface. The effects of impact load applied at different surfaces, positions of cracks’ axis and the thickness of the strips of the composite material on the possibilities of cracks’ arrest are depicted graphically for different particular cases.

Thermal Nonlinear Buckling of Shear Deformable Functionally Graded Cylindrical Shells with Porosities

In this paper, for the first time, two-term deflection is suggested for nonlinear analysis of shear deformable cylindrical shells and, to the authors’ knowledge, a first investigation on thermal nonlinear buckling behavior of functionally graded material (FGM) cylindrical shells including porosities is presented. Material properties are assumed to be temperature dependent, volume fractions of constituents are graded in the thickness direction according to a power law, and effective properties of porous FGM are estimated according to a modified rule of mixture. Porosities are evenly or unevenly distributed within the shell. Mathematical formulations are based on a first-order shear deformation theory taking into account geometrical nonlinearity and elasticity of in-plane restraints of edges. Two-term deflection and a multiterm stress function are assumed to satisfy simply supported boundary conditions. The Galerkin method and an iteration procedure are employed to evaluate critical buckling temperatures and trace temperature-deflection paths in the postbuckling region. The two-term solution of deflection is a significant suggestion for nonlinear buckling analysis of shear deformable cylindrical shells in general and thermal postbuckling analysis of porous FGM cylindrical shells in particular. This two-term solution includes inherent uniform deflection at the prebuckling state and overcomes mathematically tremendous difficulty in using three-term deflection for shear deformable cylindrical shells. Besides this innovative approach, the obtained results are of great interest for many engineering applications. Unlike the case of mechanical loads, porosities have beneficial influences on the stability of thermally loaded FGM cylindrical shells.

On the determination of first-mode stress intensity factors and T-stress in a continuous functionally graded beam using digital image correlation method

<abstract> <p>In this study, fracture parameters of epoxy/glass functionally graded composite were determined experimentally using the digital image correlation (DIC) method. A functionally graded material (FGM) with continuous variation in elastic properties was manufactured by gravity casting in vertical template. A 30% volume fraction of glass spheres was dispersed in a low viscosity resin. A single edge crack specimen was examined in a three-point bending test under mode Ⅰ loading with cracks along the gradient tendency of the material properties. The mechanical properties of FGM were calculated according to ASTM D638. The DIC technique was adopted to obtain the deformation region around the crack tip. William's series was employed to calculate stress intensity factor and T-stress. The experimental results then verified by solving the FE model using ABAQUS program. The comparison between DIC and numerical results illustrated a largely acceptable achievement.</p> </abstract>

Stochastic Extended Finite Element Implementation for Natural Frequency of Cracked Functionally Gradient and Bi-Material Structures

In this work, stochastic perturbation-based vibration characteristics of cracked bi-material and functionally graded material (FGM) domain with uncertain material properties are investigated using the extended finite element method. The level set function is implemented to track the geometrical discontinuities. The partition of unity-based extrinsic enrichment technique is employed to model the crack and material interface. The exponential law is used to model the graded material properties of FGM. The First-order perturbation technique (FOPT) is implemented to predict the standard deviation of natural frequency for the given uncertainties in the material properties. The numerical results are presented to show the effect of geometrical discontinuities and material randomness on vibration characteristics.

Sound transmission analysis of functionally graded material plates with general boundary conditions in thermal environments

A semi analytical model is proposed to perform the sound transmission analysis of functionally graded material (FGM) plates subjected to thermal loads. Temperature dependent properties of FGM and a total of three different temperature fields are taken into consideration in the present study. By means of the velocity continuity condition at the air-plate interfaces and Hamilton’s principle, the dynamic equation considering the aeroelastic coupling effect can be derived based on the first-order shear deformation theory (FSDT). Besides, the displacement components of the plate are uniformly denoted in the form of Fourier cosine series in combination with auxiliary functions for the sake of satisfying general boundary conditions. Convergent and comparative studies are conducted to validate the convergence, accuracy and versatility of the developed method. Finally, a few parametric studies with respect to the effects of various important parameters, including the boundary condition, material distribution, thermal environments and temperature dependent material properties, on the sound transmission loss (STL) through FGM plates excited by a diffuse sound field are analyzed in detailed.

Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study

In engineering structures, to having the projected structure to serve all the engineering purposes, the theory to be used during the modeling stage is also of great importance. In the present work, an analytical solution of the free vibration of the beam composed of functionally graded materials (FGMs) is presented utilizing different beam theories. The comparison of supposed beam theory for free vibration of functionally graded (FG) beam is examined. For this aim, Euler-Bernoulli, Rayleigh, Shear, and Timoshenko beam theories are employed. The functionally graded material properties are assumed to vary continuously through the thickness direction of the beam with respect to the volume fraction of constituents. The governing equations of free vibration of FG beams are derived in the frameworks of four beam theories. Resulting equations are solved versus simply supported boundary conditions, analytically. To verify the results, comparisons are carried out with the available results. Parametrical studies are performed for discussing the effects of supposed beam theory, the variation of beam characteristics, and FGM properties on the free vibration of beams. In conclusion, it is found that the interaction between FGM properties and the supposed beam theory is of significance in terms of free vibration of the beams and that different beam theories need to be used depending on the characteristics of the beam in question.

Thermal stress analysis of functionally graded solid and hollow thick-walled structures with heat generation

PurposeThis paper aims to present thermal and mechanical stresses in solid and hollow thick-walled cylinders and spheres made of functionally graded materials (FGMs) under the effect of heat generation.Design/methodology/approachConstant internal temperature and convective external conditions in hollow bodies along with internal heat generation with a combination of outer convective conditions in solid bodies are investigated individually. The effect of the heat convection coefficient on solid bodies is additionally discussed. The variation of the FGM properties in the radial direction is adapted to the Mori–Tanaka homogenization schemes, which produces irregular and two-point linear boundary value problems that are numerically solved by the pseudospectral Chebyshev method.FindingsIt has been shown that the selection of the mixtures of FGMs has to be made correctly to keep the thermal and mechanical loads acting on objects at low levels.Originality/valueIn this study, both solid and hollow functionally graded cylinders and spheres for different boundary conditions that are as their engineering applications are examined with the proposed method. The results have demonstrated that the pseudospectral Chebyshev method has high accuracy, low calculation costs and ease of application and can be easily adapted to such engineering problems.